Nevanlinna characteristic ($T(r,f)=m(r,f)+N(r,f)$)

The characteristic of a meromorphic function is the sum of its Nevanlinna proximity function and Nevanlinna integrated counting function. It treats large values and actual poles together. Replacing $f$ by $f-a$ changes the characteristic by at most $\log (1+|a|)$.

The growth order of a meromorphic function is the upper power exponent of its Nevanlinna characteristic, with constants assigned order zero. It is different from the local signed order at a zero or pole. A Nevanlinna integrated counting function has an analogous growth order. The Nevanlinna first main theorem bounds every counting order by the function order.

At most two values can have counting order smaller than a positive finite function order. Otherwise the Nevanlinna second main theorem with three such values bounds the characteristic by a smaller power outside a finite-length exceptional set. Monotonicity extends this bound to all large radii by choosing a good radius in $[r,r+1]$, contradicting the original order. Order zero is immediate from nonnegativity and the first theorem.

For a degree-$d$ rational function $U$ of the Riemann sphere, choose relatively prime homogeneous polynomials $(P,Q)$ of degree $d$. Their joint norm on the unit three-sphere has positive lower and finite upper bounds. Thus $\log \Vert (P,Q)(v)\Vert -d\log \Vert v\Vert$ is bounded. Averaging over a circle using a nonvanishing entire homogeneous lift of a meromorphic function proves the displayed law for the spherical characteristic. Jensen's formula and $0\le \frac{1}{2}\log (1+x^2)-{\log }^{+}x\le \frac{1}{2}\log 2$ show that it also holds for the ordinary Nevanlinna characteristic.